If the shortest distance between the lines L _1: r =(2+ ) i +(1-3… - Vidyadip

MCQ

If the shortest distance between the lines
$ L _1: \overrightarrow{ r }=(2+\lambda) \hat{ i }+(1-3 \lambda) \hat{ j }+(3+4 \lambda) \hat{ k }, \lambda \in R$
$L _2: \overrightarrow{ r }=2(1+\mu) \hat{ i }+3(1+\mu) \hat{ j }+(5+\mu) \hat{ k }, \mu \in R $
is $\frac{ m }{\sqrt{ n }}$, where $\operatorname{gcd}( m , n )=1$, then the value of $m + n$ equals.

A
384
B
387
C
377
D
390

✓

Answer

Shortes distance $(C D)=\left|\frac{\overrightarrow{ AB } \cdot \overrightarrow{ p } \times \overrightarrow{ q }}{|\overrightarrow{ p } \times \overrightarrow{ q }|}\right|$
$=\left|\frac{(0 \hat{ i }+2 \hat{ j }+2 \hat{ k }) \cdot(-15 \hat{ i }+7 \hat{ j }+9 \hat{ k })}{\sqrt{355}}\right|$
$=\frac{0+14+18}{\sqrt{355}}$
$=\frac{32}{\sqrt{355}}$
$\therefore m+ n$
$=32+355$
$=387$

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